Lower Bounds for (Non-monotone) Comparator Circuits

Comparator circuits are a natural circuit model for studying the concept of bounded fan-out computations, which intuitively corresponds to whether or not a computational model can make “copies” of intermediate computational steps. Comparator circuits are believed to be weaker than general Boolean circuits, but they can simulate Branching Programs and Boolean formulas. In this paper we prove the first superlinear lower bounds in the general (non-monotone) version of this model for an explicitly defined function. More precisely, we prove that the n-bit Element Distinctness function requires Ω((n/ log n)3/2) size comparator circuits. 2012 ACM Subject Classification Theory of computation → Circuit complexity

[1]  Ingo Wegener,et al.  The complexity of Boolean functions , 1987 .

[2]  Avi Wigderson,et al.  On span programs , 1993, [1993] Proceedings of the Eigth Annual Structure in Complexity Theory Conference.

[3]  Dietmar Uhlig.  Boolean Functions with a Large Number of Subfunctions and Small Complexity and Depth , 1991, FCT.

[4]  Ernst W. Mayr,et al.  The complexity of circuit value and network stability , 1989, [1989] Proceedings. Structure in Complexity Theory Fourth Annual Conference.

[5]  Ernst W. Mayr,et al.  The Complexity of Circuit Value and Network Stability , 1992, J. Comput. Syst. Sci..

[6]  Stasys Jukna,et al.  Boolean Function Complexity Advances and Frontiers , 2012, Bull. EATCS.

[7]  Avishay Tal,et al.  Cubic Formula Size Lower Bounds Based on Compositions with Majority , 2018, Electron. Colloquium Comput. Complex..

[8]  Kazuo Iwama,et al.  An Explicit Lower Bound of 5n - o(n) for Boolean Circuits , 2002, MFCS.

[9]  David Zuckerman,et al.  Mining Circuit Lower Bound Proofs for Meta-algorithms , 2014, Computational Complexity Conference.

[10]  Andrej Bogdanov.  Small bias requires large formulas , 2017, Electron. Colloquium Comput. Complex..

[11]  E. Szemerédi,et al.  O(n LOG n) SORTING NETWORK. , 1983 .

[12]  David Thomas,et al.  The Art in Computer Programming , 2001 .

[13]  V. M. Khrapchenko.  Method of determining lower bounds for the complexity of P-schemes , 1971 .

[14]  Michael T. Goodrich,et al.  Zig-zag sort: a simple deterministic data-oblivious sorting algorithm running in O(n log n) time , 2014, STOC.

[15]  Anna Gál,et al.  Lower bounds for monotone span programs , 2005, computational complexity.

[16]  Ran Raz,et al.  Improved Average-Case Lower Bounds for De Morgan Formula Size: Matching Worst-Case Lower Bound , 2017, SIAM J. Comput..

[17]  Toniann Pitassi,et al.  Exponential Lower Bounds for Monotone Span Programs , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[18]  Or Meir,et al.  Toward the KRW Composition Conjecture: Cubic Formula Lower Bounds via Communication Complexity , 2016, computational complexity.

[19]  M. V. Wilkes,et al.  The Art of Computer Programming, Volume 3, Sorting and Searching , 1974 .

[20]  Avishay Tal,et al.  Formula lower bounds via the quantum method , 2017, STOC.

[21]  Avishay Tal,et al.  Shrinkage of De Morgan Formulae by Spectral Techniques , 2014, 2014 IEEE 55th Annual Symposium on Foundations of Computer Science.

[22]  Ran Raz,et al.  Improved Average-Case Lower Bounds for DeMorgan Formula Size , 2013, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science.

[23]  Ronald L. Graham,et al.  A mathematical study of a model of magnetic domain interactions , 1970, Bell Syst. Tech. J..

[24]  Ran Raz,et al.  Average-case lower bounds for formula size , 2013, STOC '13.

[25]  M. Paterson,et al.  Boolean Function Complexity , 1992 .

[26]  Johan Hå stad.  The Shrinkage Exponent of de Morgan Formulas is 2 , 1998 .

[27]  Ernst W. Mayr,et al.  The computational complexity of the circuit value and network stability problems , 1990 .

[28]  Johan Håstad,et al.  Almost optimal lower bounds for small depth circuits , 1986, STOC '86.

[29]  Dai Tri Man Le,et al.  The complexity of the comparator circuit value problem , 2012, TOCT.

[30]  Kenneth E. Batcher,et al.  Sorting networks and their applications , 1968, AFIPS Spring Joint Computing Conference.